In this paper we regard languages and their acceptors - such as deterministic or weighted automata, transducers, or monoids - as
functors from input categories that specify the type of the
languages and of the machines to categories that specify the type of
outputs.
Our results are as follows: a) We provide sufficient conditions on the output category so that minimization of the corresponding automata is guaranteed. b) We show how to lift adjunctions between the categories for output values to adjunctions between categories of automata. c) We show how this framework can be applied to several phenomena in automata theory, starting with determinization and
minimization (previously studied from a coalgebraic and duality theoretic perspective). We apply in particular these techniques to
Choffrut\u27s minimization algorithm for subsequential transducers and revisit Brzozowski\u27s minimization algorithm
